Question 1210418
<pre>
If (3-x)+(6)+(7-5x) is a geometric series,find two possible values for
a) x
b)the common ratio
c)the sum of the Gp
pls show workings

Let the common ratio be r

Then
{{{system(
(3-x)*r = (6),
(6)*r = (7-5x))}}}

Solve the first equation for r:

{{{(3-x)*r = (6)}}}
{{{r = 6/(3-x)}}}

Substitute the right side of that for r in the second equation:

{{{(6)*r = (7-5x)}}}

{{{(6)*(6/(3-x)) = (7-5x)}}}

{{{36/(3-x) = 7-5x}}}
{{{36=(3-x)(7-5x)}}}
{{{36=21-22x+5x^2}}}
{{{15+22x-5x^2=0}}}
{{{-5x^2+22x+15=0}}}
{{{5x^2-22x-15=0}}}
{{{(5x+3)(x-5)=0}}}
5x+3=0; x-5=0
5x=-3;    x=5 
x=-3/5

Substitute x=-3/5 in 

{{{r = 6/(3-x)}}}
{{{r = 6^""/(3-(-3/5))}}}
{{{r = 6^""/(3+3/5)}}}
Multiply top and bottom by 5
{{{r = 30/(15+3)))
{{{r = 30/18}}}
{{{r = 5/3}}}

Substitute x=5 in 

{{{r = 6/(3-x)}}}
{{{r = 6/(3-5)}}}
{{{r = 6/(-2)}}}
{{{r = -3}}}

a)x = -3/5 or 5
b)the common ratio = r = 5/3 or r = -3
c)the sum of the GP: 
if x = -3/5               if x = 5
{{{(3-x)+(6)+(7-5x)}}}        ditto 
{{{3-x+6+7-5x}}}            ditto
{{{16-6x}}}                   ditto
{{{16-6(-3/5)}}}             {{{16-6(5)}}} 
{{{16+18/5}}}                 {{{16-30}}}
{{{80/5+18/5}}}                 {{{-14}}}
{{{98/5}}} 
Sum of GP = 98/5  or sum of GP = -14                 

Edwin</pre>